Lemma for prter326161. (Moved to impbidd181 in main set.mm and may be
deleted by mathbox owner, RM. --NM 15-May-2013.) (Contributed by
Rodolfo Medina, 12-Oct-2010.) (Proof modification is discouraged.)
(New usage is discouraged.)

Version of ceqsex2822 with an antecedent instead of a hypothesis.
(Use
ceqsexg2899 instead of this one. --NM 13-Aug-11)
(Contributed by Rodolfo
Medina, 19-Oct-2010.) (Proof modification is discouraged.)
(New usage is discouraged.)

Version of ceqsexv2823 with an antecedent instead of a hypothesis.
(Use
ceqsexgv2900 instead of this one. --NM 13-Aug-11)
(Contributed by Rodolfo
Medina, 19-Oct-2010.) (Proof modification is discouraged.)
(New usage is discouraged.)

Deduce equality of relations from equivalence of membership. (Moved to
eqrelrdv4781 in main set.mm and may be deleted by mathbox
owner, RM. --NM
20-Feb-2014.) (Contributed by Rodolfo Medina, 10-Oct-2010.)
(Proof modification is discouraged.) (New usage is discouraged.)

Another version of eqrelrdv4781. (Moved to eqrelrdv24784 in main set.mm
and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.)
(Contributed by Rodolfo Medina, 30-Sep-2010.)
(Proof modification is discouraged.) (New usage is discouraged.)

Pull a scalar multiplication out of a sum of vectors. EDITORIAL:
properly generalizes gsummulc215386, since every ring is a left module
over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by
Mario Carneiro, 5-May-2015.)

The inverse of a nonzero group element is not zero. (Contributed by
Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnz14534 in main set.mm and
may be deleted by mathbox owner, SO. --NM 23-May-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)

The inverse of a nonzero group element is a nonzero group element.
(Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnzcl14535
in main set.mm and may be deleted by mathbox owner, SO. --NM
23-May-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)

Any function from the subsets of a set to itself, which is extensive
(satisfies mrcssid13514), isotone (satisfies mrcss13513), and idempotent
(satisfies mrcidm13516) has a collection of fixed points which is a
Moore
collection, and itself is the closure operator for that collection.
This can be taken as an alternate definition for the closure operators.
This is the first half, ismrcd226185 is the second. (Contributed by
Stefan
O'Rear, 1-Feb-2015.)

A constant (represented without dummy variables) is an element of a
function set.

_Note: In the following development, we will be quite often quantifying
over functions and points in N-dimensional space (which are equivalent
to functions from an "index set"). Many of the following
theorems exist
to transfer standard facts about functions to elements of function
sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan
O'Rear, 5-May-2015.)